Topological orderings of weighted directed acyclic graphs

• We introduce a graph algorithm called a mark–unmark sequence for finding topological orderings of directed acyclic graphs.
• We show that if a directed acyclic graph has a non-negative mark–unmark sequence, then it has non-negative mark sequence.
• We show that it is NP-hard to decide if a directed acyclic graph has a non-negative topological ordering.

We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark–unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark–unmark sequence can also be realized by a mark sequence. We also investigate the question of whether a given weighted directed acyclic graph has a non-negative topological ordering. We show that even in the simple case when every vertex is a source or a sink the question is NP-complete.